A158954 Numerator of Hermite(n, 1/4).

1, 1, -7, -23, 145, 881, -4919, -47207, 228257, 3249505, -13184999, -273145399, 887134513, 27109092817, -65152896535, -3101371292039, 4716976292161, 401692501673153, -239816274060743, -58083536514994775, -21631462857761839, 9271734379541402161

Offset: 0

Author

N. J. A. Sloane, Nov 12 2009

Keywords

• sign
• frac

Examples

			Numerators of 1, 1/2, -7/4, -23/8, 145/16, 881/32, -4919/64, -47207/128, 228257/256, 3249505/512, ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Crossrefs

Cf. A000079 (denominators).

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), this sequence (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

Programs

Magma

[Numerator((&+[(-1)^k*Factorial(n)*(1/2)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018

Maple

A158954 := proc(n)
        orthopoly[H](n,1/4) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Mathematica

Numerator[Table[HermiteH[n,1/4],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2011 *)

PARI

a(n)=numerator(polhermite(n,1/4)) \\ Charles R Greathouse IV, Jan 29 2016

SageMath

[2^n*hermite(n, 1/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

D-finite with recurrence a(n) - a(n-1) + 8*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014

From G. C. Greubel, Jun 09 2018: (Start)

a(n) = 2^n*Hermite(n,1/4).

E.g.f.: exp(x-4*x^2).

a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/2)^(n-2k)/(k!*(n-2k)!)). (End)